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5. Systolic Adaptive Beamforming

T.J. Shepherd and J.G. McWhirter

With 27 Figures

In this chapter we address the important suQ.iect of adaptive beamforming or "null-steering" as applied to an adaptive antenna array for the purposes of noise cancellation. The chapter is not intended as a general review or tutorial discussion of the subject. It is concerned solely with the application of systolic arrays to adaptive beamforming networks based on least-squares minimization. The results described here centre upon a particular systolic array first proposed by Gentleman and Kung [5.1] for performing the QR decomposition ofa matrix in an efficient row-recursive manner. It will become clear in our subsequent development of the subject that this array provides the basic architectural component for the solution of a wide variety of signal processing problems. Most of the work described in this chapter was carried out during the last five or six years as part ofa joint research project between the Royal Signals and Radar Establishment (RSRE) and STC Technology Ltd (STL). The key results have been described in previous publications but no complete overview of the subject has been presented and many important details have not been reported to date. Since the techniques are now being developed for practical application in a number of laboratories worldwide, it seems appropriate to present a more complete discussion in this book.

The emphasis on the description of our own results is not intended to detract in any way from the significant advances made by many other researchers in this field. In particular, we should like to acknowledge the contributions made by Ling et al. [5.2], who have developed similar algorithms based upon the Modified Gram-Schmidt algorithm; Kalson and Yao [5.3], who have used Hilbert space operator methods; and Schreiber and Tang [5.4] and Sharman and Durrani [5.5], who have derived alternative least-squares systolic processing structures. (Further related contributions are listed in Sect. 5.11.) We should also like to mention the thorough exposition of Haykin [5.6], the contents of which overlap to some extent with the earlier sections of this chapter.

5.1 Adaptive Antenna Arrays

The objective of an adaptive antenna is to select a set of amplitude and phase weights with which to combine the outputs from the elements in an array so as to produce a far-field pattern that, in some sense, optimizes the reception of a

structures (commonplace in frequency hopped, spread spectrum formats) produce a requirement for adaptive systems having rapid convergence and high cancellation performance.

In recent years there has been considerable interest in the application of direct solution or "open loop" techniques to adaptive antenna processing in order to accommodate these increasing demands. In the context of adaptive antenna processing, these algorithms have the advantage of requiring only minimal input data to describe accurately the external environment and provide an antenna pattern capable of suppressing a wide dynamic range of jamming signals. Direct solution algorithms may be described most concisely by expressing the adaptive process as a least-squares minimization problem and, in effect, the least-squares algorithm defines the optimal path of adaptation.

The main problem with direct solution techniques is the large amount of computation required in comparison with gradient descent algorithms. Typically, a direct solution algorithm will require O(p2) floating point operations per sample time (where p is the number of antenna elements) as opposed to O(p) mostly fixed point operations for the digital implementation of a simple gradient descent algorithm. Until recently, it was impossible or, at best, impractical to perform the typical direct computation in real time for most practical applications. However, as a result of recent advances in VLSI circuit technology and the development of dedicated parallel processing architectures such as systolic and wavefront arrays, the use of direct solution techniques is now quite feasible even for fairly high bandwidth applications.

5.2 Systolic and Wavefront Arrays

A systolic array is an array of individual processing cells each of which has some local memory and is connected only to its neighbouring cells in the form of a regular lattice. In a typical systolic array, all the cells are identical except for a number of "special" cells which are required on the boundary in many applications. On each cycle of a simple clock, every cell receives data from its neighbouring cells and performs a specific operation on it. The resulting data are stored within the cell and passed on to neighbouring cells on the next clock cycle. As a result, each item ofdata is passed from cell to cell across the array in a specific direction and the term systolic is used to describe this rhythmical movement of data which is analogous to the pumping action of the human heart. Since each processor can only communicate with its nearest neighbours, it follows that all data must enter or leave the array through its boundary cells, the input and output being organized in a time-staggered manner for many applications.

The concept of a systolic array was first proposed by Kung and Leiserson [5.9J. In their seminal paper they showed how a number of important matrix

156 T.J. Shepherd and J.G. McWhirter

computations such as matrix-vector multiplication, matrix-matrix multiplication, and even matrix LU decomposition could be carried out in a pipelined manner using dedicated arrays of simple inner product step (multiply and accumulate) processors. In a subsequent paper, Gentleman and Kung [5.1] showed how the QR decomposition of a matrix could be implemented using a triangular systolic array of more complicated processing nodes, and most of the research described in this chapter is based on that architecture. Systolic arrays exhibit many desirable properties such as regularity and local interconnections which render them suitable for VLSI. Furthermore, the control overhead is extremely low, the only requirement being a simple globally distributed clock. However, the need to distribute a common clock signal to every processor without incurring any appreciable clock skew is one possible disadvantage of the systolic array approach, particularly in large multi-processor systems.

The wavefront array architecture proposed by Kung et al. [5.10] constitutes an important variant of the systolic array architecture for which the potential problems of clock skew are avoided. In a wavefront array processor, the required computation is distributed in exactly the same way over an array of elementary processors as it would be for the corresponding systolic array. Unlike its systolic counterpart, however, the wavefront array does not operate synchronously. Instead, the operation of each processor is controlled locally and depends on the necessary input data being available and previous outputs having been received by the appropriate neighbouring processors. Furthermore, it is not necessary to impose a temporal skew on the input data since the associated processing wavefront develops naturally within this type of self-timed array. In order to operate in the wavefront array mode, each processing element must incorporate some additional circuitry to implement a bi-directional handshake on each of its input/output links and so ensure that the necessary communication protocol is observed. This represents an overhead which is not negligible, but which can easily be absorbed within a processor of moderate complexity.

Since the mapping of any algorithm onto a wavefront array processor is identical to that of its systolic counterpart, it is sufficient for the purposes of algorithm and architecture development to consider only systolic arrays. In this chapter we show how a least-squares adaptive beamformer may be implemented in the form ofa triangular systolic array. This development clearly illustrates the benefit of matching algorithms and architectures for digital signal processing. It is obviously important for any application to choose an algorithm that solves the required problem in a numerically stable and well-conditioned manner. However, it is also important to ensure that the algorithm can be implemented at the required data rate using an efficient, cost-effective processor. The need to take both aspects into account simultaneously provides the signal processing designer with a very considerable challenge. Fortunately, as our research has shown, the different requirements need not be mutually opposed and, indeed,

158 T.J. Shepherd and J.G. McWhirter

5.3.1 Canonical Configuration

Consider the form of adaptive combiner which is illustrated in Fig. 5.2. The inputs to the combiner take the form of a primary signal y(t) and set of p - 1 (complex) auxiliary signals denoted here by the vector x(t). The weight w is adjusted to minimize the power of the combined output signal which is given at time t by

where the superscript T denotes matrix transposition. This type of adaptive linear combiner may be used in a wide range of adaptive antenna applications.

It is well known, for example, how it may be applied to adaptive sidelobe cancellation [5.7]. In this case the prifuary signal constitutes the output from a main (high gain) antenna while the auxiliary signals are obtained from an array of p - 1 auxiliary antennae. The adaptive combiner serves to modify the beam pattern of the overall antenna system by directing deep nulls towards jamming waveforms received via the sidelobes of the main antenna.

It is also well known how this form of adaptive combiner may be used in conjunction with a suitable reference signal to control a more general antenna array in which all the elements are essentially equivalent [5.8]. The reference signal, which is assumed to be correlated with the desired signal, provides the primary input to the combiner while the signals received by the antenna array provide the p - 1 auxiliary inputs. In this case the weighted sum ofthe auxiliary inputs provides as close a match as possible to the reference signal and hence produces the desired output from the beamformer.

The basic combiner illustrated in Fig. 5.2 may also be used in the so-called "power inversion" mode which has particular application to communications (for a discussion see, for example, Hudson [5.11]). In this case the p antenna

.!...(t) y(t)

ARRAY OUTPUT e(t)

Fig. 5.2. Canonical configuration of adaptive combiner

elements are assumed to be omni-directional and of comparable gain. The received signals are fed into the combiner, one of them going to the primary channel and thus having its weight coefficient constrained to unity. The other p - 1 signals enter the auxiliary channels with their adaptive weights initialized to zero and so, prior to adaptation, the overall beam pattern is determined solely by the (omni-directional) response of the primary element. This "end-element- clamped" configuration provides no inherent mechanism to inhibit the adaptive process from nulling the desired signal. However, the system is only alldwed to adapt when the desired signal is known to be absent. When it is present, the weight vector is frozen, thus allowing signal reception. This is referred to as the power inversion mode of operation because the differential interference powers received by the antenna elements are inverted by the combiner.

A particularly important application of adaptive antenna arrays requires the power of a p-element combined signal

(5.2)

to be minimized subject to a linear beam constraint of the form

(5.3)

This constraint ensures that the gain of the antenna array maintains a constant value J.l in a given "look direction" specified by the vector c. It is worth pointing out that the end-element-clamped configuration described above constitutes a particularly simple form of linearly constrained process in which the constraint

together with a gain J.l of unity. However, the incorporation of a general linear constraint is not so straightforward. A number of techniques have been proposed in the literature but in all cases the resulting implementation is extremely cumbersome. For example, Widrow et al. [5.8] suggested the injection of an artificial look-direction signal into the antenna array receiver channels and introducing a corresponding reference signal into the adaptive process. This technique then requires an additional "slave processor" to apply the adapted weight vector. Frost [5.12] also showed how a general linear constraint could be incorporated into the adaptive process using projection operator techniques; the resulting algorithm is, however, rather expensive in terms of computation.

In Sect. 5.7 of this chapter it will be shown how a single constraint may be imposed very simply using a linear pre-processor in conjunction with an end- element-clamped beamformer. The characteristics of the pre-processor are not unique, but must satisfy a simple set of conditions. It is thus possible to choose examples which lead to a particularly efficient implementation. The pre-pro- cessor concept is then generalized to execute the imposition of more than one constraint, so that the antenna array maintains a fixed gain in several directions

160 T.J. Shepherd and J.G. McWhirter

simultaneously. An efficient method will also be described whereby a single constraint may be included after the power minimization has been performed, leading trivially to the ability to solve the single constrained problem many times in parallel. This type of constraint post-processor enables the array to be employed efficiently in an effective target location and tracking mode.

From the discussion in this section it should be clear that the type of adaptive combiner illustrated in Fig. 5.2, previously referred to as the end- element-clamped configuration, has a wide range of applications in adaptive beamforming. Since it may be applied to the problem of adaptive beamforming with general linear constraints, as well as to the more obvious sidelobe cancellation problem, this will be referred to as the "canonical" configuration. We now direct our attention to the development of a novel algorithm and corresponding processor architecture which applies specifically to the canonical adaptive combiner.

5.3.2 Least-Squares Formulation

The function of the adaptive combiner in Fig. 5.2 may be defined mathematically in terms of least-squares estimation. We denote the combined array output at time tj by

(5.5)

where x(tj ) is the (p - I)-element vector of (complex) auxiliary signals at time tj and yeti) is the corresponding sample of the (complex) primary signal. We seek to minimize the residual signal power at time tn as estimated by the quantity

E2(n), where

(5.6)

For the sake of generality this unnormalized estimator includes a simple "forget factor" f3 (0 < f3 ~ 1) which generates an exponential time window and localizes the averaging procedure. Introducing a more compact matrix notation, the estimator defined in (5.6) may be expressed in the form

where